We prove an upper bound on the Wassertein distance between normalized martingales and the standard normal random variable, which extends a result of Röllin (2018). The proof is based on a method of Bolthausen (1982).

This note introduces an asymptotically normal statistic for the value function of a convex stochastic minimization programme, which may have more than one minimizer. The statistic uses a recursive estimator, based on the proximal algorithm, of one of the minimizers.

Strong laws of large numbers are obtained for non-homogeneous arbitrary dependent sequences.

An optimal design should minimum the confounding among lower-order effects. Based on an aliased component-number pattern (ACNP), a general minimum lower-order confounding (GMC) criterion is proposed to choose s-level regular designs. The paper mainly studies some properties of s-level designs in terms of complementary sets. Further, an sn−m design has GMC only if its complementary set is contained

We consider a Stein’s approach to estimate a covariance matrix using regularization of the sample covariance matrix Cholesky factor. We propose a method to estimate accurately the regularization vector which minimizes the risk associated with the recently introduced log-Cholesky metric.

In the present study, the identifiability problem is considered for a general form of the joint model for two types of responses. The presented three theorems make it easier to verify identifiability. The ideas or techniques from the proofs can be used to extend the work to other joint models.

Operator fractional Brownian motion (OFBM) is a multivariate extension of fractional Brownian motion and has operator self-similarity. The dependence structure across the components of OFBM is determined by the Hurst matrix H and E(XH(1)XH(1)′). In this paper, the estimators of H with wavelet method is compared in continuous sample path and discrete sample path. It is proved that with a discrete sample

Hall and Marron (1987) introduced kernel estimators of integrals of the squared m-order derivatives of a probability density. Mokkadem and Pelletier (2020) gave recursive versions of their estimators, but the main drawback of these estimators is that their update requires the use of all past data. The aim of this paper is the study of online versions of the estimators introduced by Hall and Marron

Maximin distance Latin hypercube designs (LHDs) are frequently used in computer experiments, but their constructions are challenging. In this paper, we present some new results connecting maximin L2-distance optimality and near orthogonality for mirror-symmetric LHDs. We further propose a simple and effective method for constructing nearly orthogonal LHDs that can yield almost the largest minimum distance

In this paper, a nonparametric estimator for the regression function is constructed when the covariates are contaminated with the multivariate Laplace measurement error. The proposed estimator is based upon a simple relationship between the regression function and the conditional expectation of the regression function given the proxy data, as well as the second derivative of this expectation. Large

Notions of depth in regression have been introduced and studied in the literature. Regression depth (RD) of Rousseeuw and Hubert (1999), the most famous one, is a direct extension of Tukey location depth (Tukey, 1975) to regression. Like its location counterpart, the most remarkable advantage of the notion of depth in regression is to directly introduce the maximum (or deepest) regression depth estimator

This note presents relationships between a base copula and the copula induced by distortion of the base copula in the strength of tail dependence measured by tail dependence coefficients and tail orders. We derive a theorem that determines the relationships for distortion functions satisfying conditions related to regular variation. In addition to the well-known logarithmic, power and dual-power distortions

We consider the last zero crossing time Tμ,t of a Brownian motion, with drift μ≠0, in the time interval [0,t]. We prove the large deviation principle of {Tμr,t:r>0} as r tends to infinity. Moreover, motivated by the results on moderate deviations in the literature, we also prove a class of large deviation principles for the same random variables with different scalings, which are governed by the same

Multivariate subordinated Lévy processes are widely employed in finance for modeling multivariate asset returns. We propose to exploit non-linear dependence among financial assets through multivariate cumulants of these processes, for which we provide a closed form formula by using the multi-index generalized Bell polynomials. Using multivariate cumulants, we perform a sensitivity analysis, to investigate

This article studies asymptotic approximations of ruin probabilities of multivariate random walks with heavy-tailed increments. Under our assumptions, the distributions of the increments are closely connected to multivariate subexponentiality and admit dependence between components.

Split-plot designs are very popular in agricultural experiments where two factors require different sizes of plots. In traditional split-plot designs, main plot treatments are allocated using a randomized complete block design and subplot treatments are allocated within each main pot. However, often it may not be feasible to accommodate all the subplot treatments in each main plot. The purpose of this

Let XH={XH(s),s∈RN1} and XK={XK(t),t∈RN2} be two independent centered space–time anisotropic Gaussian random fields taking values in Rd. In this paper, we study the existence of intersections of XH and XK. Furthermore, we determine the Hausdorff dimensions of the set of intersection times and the set of intersection points of the random fields, respectively. Our results generalize the corresponding

The Weibull distribution is one of the most used tools in reliability analysis. In this paper, assuming a Bayesian approach, we propose necessary and sufficient conditions to verify when improper priors lead to proper posteriors for the parameters of the Weibull distribution in the presence of complete or right-censored data. Additionally, we proposed sufficient conditions to verify if the obtained

This paper proposes an asymptotically distribution free test for fitting a parametric model to the autoregressive function in the AR(1) model in the presence of measurement error. The test is based on a martingale transform of a certain marked residual empirical process. A simulation study assessing the finite sample level and power performance of the proposed test is also included.

Methods for the construction of hypothesis tests based on multiple data splitting are presented. The tests combine p-values, and exhibit overall Type 1 error control. But, it is also shown that multiple data splitting may have worse power than single splitting.

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of band Toeplitz matrices with independent Brownian motion entries, as the dimension of matrix goes to infinity. Here the band width bn of the n×n Toeplitz matrix goes to infinity with n at the rate of o(n).

In this short paper, we are concerned with the blowup phenomenon of stochastic parabolic equations. By using the comparison principle and the results for deterministic parabolic equations, we obtain blowup results of solutions for stochastic parabolic equations.

The moment of a positive random variable can be obtained by using the survival function. In this paper, we extend existing studies by giving formulae of the expectation of a more general function with respect to a random variable by using either cumulative distribution function or survival function. Both univariate and multivariate examples are given. These formulae can evaluate the expectation when

In this note we present new perturbation bounds for the stationary distribution of Markov chains with denumerable state space. Bounds are obtained using the Dobrushin’s coefficient δ(P) and the ergodic coefficient α(P). Bounds are shown to be relevant when α(P)<δ(P). In addition we present bounds only involving the coefficient α(P).

Strong approximations of uniform transport processes to the standard Brownian motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego, 1971) and improves their rate of strong

We develop an algorithm that makes it possible to generate all correlation matrices satisfying a constraint on their average value. We extend the results to the case of multiple constraints. These results can be used to assess the extent to which methodologies driven by correlation matrices are robust to misspecification thereof.

Convergence rate to the stationary distribution for continuous-time Markov processes can be studied using Lyapunov functions. Recent work by the author provided explicit rates of convergence in special case of a reflected jump–diffusion on a half-line. These results are proved for total variation distance and its generalizations: measure distances defined by test functions regardless of their continuity

The regression dependence is a practical dependent structure which provides a good mechanism to describe non-life insurance businesses, and further allows applications in various areas. In this paper, we investigate large deviations for random sums of extended negatively dependent random variables in the general renewal risk model with the regression size-dependent structure. For heavy-tailed distributions

We investigate certain positive random variables having moments of Gamma type. We give some necessary conditions and some sufficient conditions for their existence. In particular, we observe that the Weber–Schafheitlin formula for the Bessel function allows one to construct non-trivial moments of Gamma type having a signed spectral measure.

In the study of nontransitive dice, seemingly paradoxically, the probability that one die rolls higher than another is not transitive. We prove a more general result which implies that a cyclic sum of ordering probabilities for n random variables can be much greater than 1, and that each probability can be much greater than 1n. In particular, we establish an upper bound for this sum and give necessary

Consider a non-explosive positive Feller process with no negative jumps. It is shown in this note that when infinity is an entrance boundary, in the sense that the entrance times of the process remain bounded when the initial value tends to infinity, the process admits a Feller extension on the compactified state space [0,∞]. Moreover, when started from infinity, the extended Markov process on [0,∞]

Let b(x) be the probability that a sum of independent Bernoulli random variables with parameters p1,p2,p3,…∈[0,1) equals x, where λ≔p1+p2+p3+⋯ is finite. We prove two inequalities for the maximum of the density ratio b(x)∕πλ(x), where πλ is the probability mass function of the Poisson distribution with parameter λ.

Data sharpening can reduce bias in non-parametric regression and density estimation. Firth’s (1993) approach to bias reduction through adjustment of the score function provides an underlying framework for data sharpening and extensions, such as sharpened derivative estimation in kernel regression.

Computer experiments with both quantitative and qualitative factors are commonly encountered in practice. Several literatures found that if the cross-correlation between an auxiliary response and the target response (i.e., the response to be predicted) is small, the information of such an auxiliary response may reduce the prediction accuracy of the target response. In this work, we use the prediction

A consistent test for detecting DMTTF property is proposed. The exact null distribution of the test statistic as well as its asymptotic distribution have been derived. Efficacy values are calculated, and a simulation study is carried out to assess the performance of the test. Finally, the test is applied to some real-life data sets.

It is shown that when X follows a location-scale distribution with continuous density function that has support R and 01 and all values of the location and scale parameters. An explicit expression for the minimum λ is proven for the family of scaled and shifted t-distributions.

The maximum likelihood estimator (MLE) under the parametric set-up with the right-censored (RC) data rarely has a closed form solution. The exponential distribution is an exception. Moreover, the parametric MLE’s with RC regression data are currently computed by iterative algorithms. We show that the MLE of the exponential distribution under the linear regression model with RC data has a closed form

Sticky Brownian motion on the real line can be obtained as a weak solution of a system of stochastic differential equations. We find the conditional distribution of the process given the driving Brownian motion, both at an independent exponential time and at a fixed time t>0. As a classical problem, we find the distribution of the occupation times of a half-line, and at 0, which is the sticky point

We prove two propositions with conditions that a system, which is described by a transient Markov chain, will display local stability. Examples of such systems include partly overloaded Jackson networks, partly overloaded polling systems, and overloaded multi-server queues with skill based service, under first come first served policy.

We derive the Hammersley–Chapman–Robbins inequality for discrete quantum parameter models in the presence of time dependent measurements. The extension determines a discrete counterpart of the classical Fisher information. We provide an illustration concerning a quantum optics problem.

With a bound on cost and bounds on stratum sample sizes, a cost weighted objective function is decomposed to reveal a general simple exact optimal sample allocation algorithm for minimizing sampling variances. Some known allocation methods are special cases.

Consider testing multiple hypotheses in the setting where the p-values of all hypotheses are unknown and thus have to be approximated using Monte Carlo simulations. One class of algorithms published in the literature for this scenario provides guarantees on the correctness of their testing result through the computation of confidence statements on all approximated p-values. This article focuses on

In this paper, we propose a new approach of innovated scalable dynamic learning (ISDL) for estimating time-varying graphical structures. Motivated by the innovated transformation, we convert the original problem into large covariance matrix estimation and exploit the scaled Lasso with kernel smoothing to simplify the tuning procedure. In addition, we show that our method has theoretical guarantees

Narayana numbers appear in many places in combinatorics and probability, and it is known that they are asymptotically normal. Using Stein’s method of exchangeable pairs, we provide an error of approximation in total variation to a symmetric binomial distribution of order n−1, which also implies a Kolmogorov bound of order n−1∕2 for the normal approximation. Our exchangeable pair is based on a birth–death

There is a widely known intriguing phenomenon that discrete-time GARCH and stochastic volatility (SV) models share the same continuous-time diffusion model as their weak convergence limit, but statistically, the GARCH model is not asymptotically equivalent to the SV or diffusion model. This paper investigates GARCH-type quasi-likelihood ratios for the SV and diffusion models whose own likelihoods are

Stochastic orders are mathematical methods allowing the comparison of random quantities. Probably the most usual one is stochastic dominance, which is based on the comparison of univariate cumulative distribution functions. Although it has been commonly applied, it does not consider the dependence between the random variables. This paper introduces a new stochastic order that slightly modifies stochastic

Matrix Factorization is a widely used technique for modeling pairwise and matrix-like data. It is frequently used in pattern recognition, topic analysis and other areas. Side information is often available, however utilization of this additional information is problematic in the pure matrix factorization framework. This article proposes a novel method of utilizing side information by combining arbitrary

We give a two-dimensional central limit theorem (CLT) for the second-order quadratic variation of the centered Gaussian processes on [0,T]. Though the approach we use is well known in the literature, the conditions under which the CLT holds are usually based on differentiability of the corresponding covariance function. In our case, we replace differentiability conditions by the convergence of the

We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standard geodesic distance. The radial process is closely related to a Wright–Fisher diffusion, increments of which can be simulated exactly using the recent work of Jenkins & Spanò (2017). The rapid spinning phenomenon

In this paper we revisit the integral functional of geometric Brownian motion It=∫0te−(μs+σWs)ds, where μ∈R, σ>0 and (Ws)s>0 is a standard Brownian motion. Specifically, we calculate the Laplace transform in t of the cumulative distribution function and of the probability density function of this functional.

In this note we introduce a notion of optional Snell envelopes over the so-called split stopping times. As application, we give a solution to reflected backward stochastic differential equations when the barrier is an optional positive process neither right-continuous nor left-continuous.

In this paper, jackknife empirical likelihood is proposed to be applied in stationary time series models. By applying the jackknife method to Whittle estimator, we obtain new asymptotically independent pseudo samples which will be used to construct linear constraints for empirical likelihood. The jackknife empirical log-likelihood ratio is shown to follow a chi-square limiting distribution, which validates

A closed-form expression for the remaining life expectancy based on the gamma-Gompertz model has been derived recently in the literature. However, the closed-form expression available to users is not correct. We provide, therefore, a valid and correct expression for the life expectancy at age x by using elementary calculus. We also consider a comparative numerical study by using real data to confirm

We prove the consistency of the ℓ1 penalized negative binomial regression (NBR). A real data application about German health care demand shows that the ℓ1 penalized NBR produces a more concise but more accurate model, comparing to the classical NBR.

Kotlarski (1967) establishes a fundamental result on identification of marginal distributions of independent random variables X, Y, and Z from the joint distribution of random variables (U,V), where (U,V)=(X+Z,Y+Z). We extend this result to the case (U,V)=(X+aZ1+bZ2,Y+cZ1+dZ2), where Z1 and Z2 are identically distributed, and a, b, c, and d are different weights. As an outgrowth of the proof, we also

In this article, we propose model-free marginal and conditional feature screening approaches for ultra-high dimensional competing risks data. The suggested procedures possess ranking consistency and sure screening properties. Their finite-sample performances are verified through numerical studies.

Independently scattered random measures are usually defined and constructed under the additional condition of being infinitely divisible, a rather strong condition. A more natural condition is the assumption that the random measure has no atoms, that is being atomless. We show that atomless random measures on δ-rings are necessarily infinitely divisible, so infinite divisibility is in fact a quite

A new measure of dispersion is presented here that generalizes entropy for positive data. It is intrinsically linked to a measure of central tendency and is determined by the data through a power transformation that best symmetrizes the observations.

When functional data are observed on parts of the domain, it is of interest to recover the missing parts of curves. Kraus (2015) proposed a linear reconstruction method based on ridge regularization. Kneip and Liebl (2019) argue that an assumption under which Kraus (2015) established the consistency of the ridge method is too restrictive and propose a principal component reconstruction method that

We study the properties of an M-estimator arising from the minimization of an integrated version of the quantile loss function. The estimator depends on a tuning parameter which controls the degree of robustness. We show that the sample median and the sample mean are obtained as limit cases. Consistency and asymptotic normality are established and a link with the Hodges–Lehmann estimator and the Wilcoxon